Entry
Problem solving: Method: Overview: Can you give an overview of possible problem solving methods?
May 2nd, 2008 07:34
Knud van Eeden, Sakir Ali,
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--- Knud van Eeden --- 12 October 2020 - 05:16 pm --------------------
Problem solving: Method: Overview: Can you give an overview of
possible problem solving methods?
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Looked for is some collection of general methods which you could use to
solve a variety of problems in a wide variety of fields (computer
science, programming, debugging, fixing hardware, mathematics, physics,
chemistry, mechanics, sports, natural languages, learning, ...)
---
To solve a problem, you use usually a combination of:
1. 'Deterministic methods', or thus methods which for sure guarantee a
solution
or
2. 'non deterministic methods', or thus methods which do not
guarantee a solution
For example because
1. there are simply too many possible solutions which have to
be tried
2. too little data is known
3. too little knowledge about how the problem should be solved
is present
1. The heuristic (or 'Rules of thumb', it might work but it also
might not work) method
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When solving a problem you might use zero or more of the below methods
and processes (or combinations and overlap of it):
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---
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1. Association:
1. Analogy:
Relationship between two or more similar problems or concepts,
bringing the two or more concepts into some kind of alignment or
relationship that highlights their similarities and differences.
It involves a mental mapping between two or more disciplines.
1. Transfer of information
1. Analogical transfer
1. Selective Comparison.
Selective comparison involves relating newly acquired
information to information acquired in the past.
Selective encoding and selective combination frequently
generate the need to retrieve old information. In a
learning situation a problem solution may begin with
either new knowledge or old knowledge.
1.'Can I find a similar solution in
another system'
2. Historic analogical transfer
Prior knowledge reduces the search space of the problem to
the most important aspects. It automatees some of the
steps, opens up space in working memory to solve problem.
1. 'Can I relate the problem to a similar problem which I
know how to solve and solved before'
2. 'It is more likely I can solve this difficult problem
if I have just been told how to solve a similar one'
3. Isomorphic Problems
Two problems are isomorphic if their formal structure is
the same, and only their content differs
2. Solve a specific example
1. 'After I solved that example, I saw quite good what to do
next'
---
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2. Assumption:
1. Accept or reject solutions
1. Guessing solutions
1. Trial and error
1. Phantasy
1. 'Could I just imagine any kind of solution'
1. Brain storming
1. 'Even the wildest ideas are welcome'
2. Create not yet known series of patterns
and solutions
2. Probability
1. 'If I try I have always a chance to have it right'
2. Serendipity
'I might by incident stumble on a solution'
2. non trial and error
1. Educated guessing
1. 'It has shown to work before elsewhere,
so it might also work here'
2. Guestimation
1. 'If I can guess an approximative solution,
it will help me a lot during the solving'
2. Guess the solution and then check the answer
1. 'After I guessed some solution it was easy
to show the real solution was not in that
range'
2. Compare with common sense
1. 'Does this solution makes sense'
2. 'Does this solution falls within known
limits'
2. Gedanken experiment
'Can I build some mental model, to test further'
2. Using strategies which are known to lead to a solution
2. Non guessing solutions
1. Empirical solutions
1. 'Select a solution that has shown to work before'
2. Reasoned solutions
---
---
3. Automation
1. Store the solution of your problems, so that you can use
it another time much more quicker
1. Computer
1. Create a computer program that automates the step
2. Creat a computer wizard program
---
---
4. Communication:
1. Communication with others:
1. Expert (=other long term memory)
1. 'That person must know for sure the solution'
2. non Expert (=other long term memory)
1. 'That person probably knows some solution'
3. Listen to others (=other long term memory)
1. Sometimes half a word is enough
1. 'Now I suddenly understand what to do, after
listening'
2. Ask questions about the problem.
1. 'A good listener needs only half a word'
4. Explain to others
1. 'After I explained the problem to her, I understood
it much better suddenly'
5. International open source projects
1. 'If I do not know it, then certainly somebody else.
And together we create the best solutions, with
the possible cross fertilization of all this different
disciplines'
2. Communication with yourself
1. Use subvisualization
1. Create dynamic inner visual representations using
your mental eye ('like a movie')
2. Use subvocalization
1. Talk to yourself innerly to guide your actions
using your mental voice and ear ('talk internally')
---
---
5. Creative
Creativity is a novel and unexpected way of defining or solving a
problem which leads the observer to ask, "How did you think of
that?"
1. Open minded
1. Keeping an open mind
1. 'I am open for all solutions'
1. Use all possible solution paths possible
2. Varying of possible solution methods
1. 'Could I use this method possibly for other purposes'
2. 'I will now try the new method, and not stick with my
old method'
3. Ask for many solutions
1. 'Can I think of a lot of other solutions for this
problem'
4. Use lateral thinking
'seeking to solve problems by unorthodox or apparently
illogical methods'
5. Use divergent thinking
1. 'I will try to generate a diverse assortment of
possible alternative solutions to a problem'
5. Have many ideas. The more ideas, the more likely that one
will be good
6. Expose yourself to different milieus
'I will also try this method, from that other discipline,
though I have never tried that discipline before'
7. Probable insight solution versus routine solution
2. not keeping an open mind
1. Being doubtful
1. 'That solution might never work (but it does in fact)'
2. not being doubtful
1. 'That solution will never work (but it does in fact)'
3. Ask what the hidden assumptions are or what you have
forgotten to use.
1. Novice problem solvers often limit their solutions by
assuming constraints which are not part of the problem
4. Probable routine solution versus insight solution
5. Functional Fixedness
1. 'It is not possible to use that in that way to achieve
that goal, I never used it like that before'
6. Blockages
1. Cultural
2. Ethical
3. Juridical
4. Social
5. Other
2. Questions
1. Asking the right questions
1. Input
1. Selective encoding.
Selective encoding requires sifting
out relevant from irrelevant information.
A useful tool for selective encoding, as used in
problem solving, is answering the questions:
The answer to "What is ASKED FOR?" establishes the
problem space identified in cognitive research (Newell
1990)
1. 'What is asked for'
2. 'What is given'
1. 'What data do I need to know'
3. 'What do I need to know'
2. Determining the end of your search paths
1. 'What goal do I have to achieve'
3. 'How can I achieve that goal'
The answer to "How would I FIND OUT?" and continuing to
work in left-right, top-down, or right-left, bottom-up
fashion implements the problem search and knowledge search
path in order to traverse the problem space and reach a
solution reliably and efficiently.
1. 'Which paths to choose for possible solutions'
4. 'What concepts, formulas, and rules did I apply'
5. 'What methods did I use'
6. 'How did I begin'
7. 'Have I seen this problem before'
8. 'Is it similar or dissimilar to other problems I have
done'
9. 'Could this problem be worked another way'
10. 'Can I simplify what I did'
2. not asking the right questions
---
---
6. Direction:
1. Order
1. Concentrate on the parts of the problem that can be solved.
'I know that very often parts that seem unsolvable become
solvable when other parts of the problem have been solved,
and it boosts also my confidence'
1. Start with the easier parts of the problem
2. Vary the order in which you solve the parts of the solution
1. Working forwards
1. Start with the question and work forwards towards the
goal
1. 'I know the data. How can I work out this data.
How can I combine it to get the answer'
2. 'What could I do with this given data, to achieve
the solution'
2. Working backwards
1. Start with the goal and work backwards to the question
(so combination of analysis and synthesis)
1. Determine which input you minimally need to know for
a solution of the goal
1. 'What do I need to know or do minimally in order
to get the answer or the goal fulfilled'
3. Switch between working forward and backward.
1. Although experts work forward on simple problems, they
alternate working forward and backward on difficult
problems.
---
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7. Effort
1. Priority
1. Spend much time
1. 'Now I will work on the problem until it is solved'
2. Finding solutions is 99% transpiration and 1% inspiration
1. Endurance
2. not endurance
2. non priority
---
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8. Environment:
1. Create environment adapted to solve problems
1. Location
1. 'That mountain view really inspires me'
2. 'That fresh air gives me new inspiration'
1. Room
1. 'In that room I can really solve the problem very good'
2. Book
3. Food
1. Creatine (=memory)
2. Coffee (=stimulation, memory, problem solving)
3. Peanuts (=brain stimulation)
4. Grapes (Resveratrol) (=keeps you fit)
---
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9. Feedback:
1. Change and check
1. Search location of the problem
1. 'I change something there, view the result,
and so know at least where the location of the problem
is'
2. Look back at subinformation
3. Check the subresults
4. Evaluation
1. Get the facts and be sure there actually is a problem.
1. 'Is it really a problem'
1. 'If it is not broke, then do not fix it'
---
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10. Incubation
1. Wait some time before trying to solve the problem
1. Wait for some feedback from the subconscious
(the subconscious can be seen here as a combinatorial
machine, which relentlessly examines all possible problem
solutions and combinations, mostly during sleep, and
supplies you then suddenly with answers sometimes)
1. During sleeping your brain is very active combining
and trying solutions
1. 'After a few nights sleep a solution might appear'
2. Aha erlebnis
1. 'Aha, now I see the solution suddenly in front of my
mental eye'
1. Solution 'like a movie'
2. Solution not 'like a movie'
2. Collect as much information possible
1. Read books
2. Check the Internet
3. Talk to others
4. A solution likes a prepared mind
The benefits of incubation can be enhanced in two ways:
1. Invest enough time in the problem initially
2. Allow sufficient time for incubation to permit the
reorganization of information
3. Parallelism
1. 'Somewhere else on the world they might been working on it
I just wait and see'
2. Do not wait some time before trying to solve the probem
1. 'I try to solve the problem immediately, anyhow'
1. Workaround
1. 'Can I solve the problem a bit at least temporarily'
---
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11. Logic
1. Deductive reasoning
1. Impossible for the conclusion to be false if the
premises are true
1. First order logic
1. Syllogism or Modus ponens
1. if A then B ... given A then infer B
1. A = If it is raining I take an umbrella
B = It is raining
A + B = Therefore I take an umbrella
2. Modus tollens
1 if A then B ... if A is false infer B is false
3. Affirmation of consequent
4. Denial of antecedent
---
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12. Model
Follow systematic, known plans to achieve your goals
1. Use a known model
1. Polya model
1. Visualize: Try to draw a picture
2. Data: Ask what is known
1. 'What are your data'
3. Goal: Ask what has to be the goal
1. 'What do I want to know'
4. Means: Ask how it can be achieved
1. 'How could I achieve that'
2. Herbert Simon model
1. A problem has initial state and goal state
2. number of intermediate states and alternative paths
3. mental operators used to move from one state to the next
taking account of legal constraints
4 knowledge and heuristics to search space
5 limitations of cognitive system include capacity of working
memory and the speed of retrieval from long term storage
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Characteristics of problems:
1. PROBLEM SPACE (all possible configurations)
2. PROBLEM STATE (the particular configuration)
3. Key to solving a problem is to choose the right OPERATORS
(processes applied to change the configuration)
4. Problem solving is a search process: Each action takes us
from one part of the problem space to another
Two different kinds of problems that you have to tackle:
1. Ill-defined problems
You do not know exactly what the goal is or how to
approach the problem.
1. 'I have to write a book about world peace'
2. Well-defined problems
The goal is clear, the beginning state is clear, you
know what is involved
1. 'I have to put together that puzzle'
3. State Space Search
1. states:
1. initial state
2. goal(s)
3. intermediate operators
4. Search Spaces (=model of the search process)
1. use trees and graphs
1. represent states as nodes
2. represent operators as arcs
2. use simple algorithms
1. breadth first search
2. depth first search
1. Means-ends analysis
1. Combination of forward and backward
strategies Continuously set new
subgoals to shorten the distance
between the current state and the
final goal.
2. Hill climbing
3. Woods Model:
1. Engage/Motivation
2. Define
3. Explore
4. Plan
5. Do it
6. Check
7. Evaluate/reflect
2. not use a known model
1. Simulation
1. Using the computer
1. program which mimics the problem
2. not using the computer
1. Build a maquette of your problem
2. Recurring similar patterns
1. Physics problems frequently involve identifying the
appropriate principles and formulas based on the given
input data
---
Physics Problem Solving
1. Visualize the problem.
1. draw a sketch
2. identify the known variables and constraints
3. restate the question
4. identify the general approach to the problem
2. Describe the problem in physics terms.
1. use identified principles to construct idealized
diagram
2. symbolically specify the relevant known variables
3. symbolically specify the target variable
3. Plan a solution.
1. start with the identified physics concepts and
principles in equation form
2. apply the principles systematically to each type of
object or interaction
3. add equations of restraint that specify any special
conditions
4. work backward from the target variable until you have
determined that there is enough information to solve
the problems
5. specify the mathematical steps to solve the problem
4. Execute the plan.
1. use the rules of algebra to obtain an expression for
the desired unknown variable
2. instantiate the equation with specific values to
obtain a solution
3. solve the equation for the desired unknown
5. Check and evaluate.
1. check - is the solution complete?
2. check - is the sign of the solution correct?
3. check - does the solution have the correct units?
4. check - is the magnitude of the answer reasonable?
2. Algebra problems require an algebraic representation of
the given information followed by manipulating equations
and axioms until the answer is derived
3. Geometry problems are similar, but usually involve
graphical representations of the situation
4. Computer programming problems:
1. State the problem clearly and concisely (simple
flowcharts can be effectively used here)
2. Describe the
1. given input information
1. desired output information
3. Define a set of tests,
(including trivial cases)
4. Draw a picture of a simple situation
5. Use pseudocode
(write the steps in your own natural language,
e.g. English)
1. Write each step in your own natural language
(e.g. English)
6. Work the problem by hand for simple cases
7. Generalize where appropriate
8. Code a solution by expanding the outline,
and or replacing your natural language
by the keywords and other structures of
the computer language you use (e.g. C#, Java,
C++, Delphi, ...)
9. Evaluate the code for all the test cases
---
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13. Motivation
1. Intrinsic motivation
1. Motivation which comes from inside of you
1. Optimistic view
1. 'I am able solve to that problem, so I will'
2. Extrinsic motivation
---
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14. Memory
1. Optimally use
Good expert problem solvers also often have
a good working memory
1. Optimal use of the senses
1. Visual
1. 'To me a visual example might be worth 1000 words'
2. non visual
1. Audio
1. For some, some music might be stimulation
2. For some, some music might not be stimulation
2. Non audio
1. Optimal use of your short term memory
1. Rehearse
2. Optimal use of your long term memory
2. Organization
1. Store subinformation
1. Use description (diaries)
1. Use the appropriate means
1. Paper (=long term memory)
2. not paper
1. Computer (=long term memory)
1. Write a computer program which stores your
solution path for reuse later
---
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15. Reduction:
1. Remove by elimination:
1. Replace with a working system
1. 'I know for sure that works elsewhere, so if it does
not work here, then that can not be the cause'
2. not replace with a working system
1. Remove by contradiction:
1. Assume the converse and look what follows from this
2. Analysis (=taking apart)
The process of solving problems is based on Analysis (taking
apart) and Synthesis (putting together). You analyze the
problem by taking it apart and identifying its components, then
you put those components back together with additional
components or in a different arrangement to synthesize the
answer. Simple problems usually involve rearranging the
components. In complex problems, you must identify the
additional components or information that is needed, and you
must be able to find it.
1. Fractionation:
1. Split the problem in parts
1. 'Try to split the problem in subproblems'
1. 'Do I see some rather independent parts in the
whole'
2. 'Do I see some parts which look easier to solve'
3. Simplification:
1. 'Try to find the simplest case'
1. When working with formulas, try substituting in numbers
first
2. 'Take away as much of the problem as possible,
until only the bare bones are left'
1. 'Try to find a simple example'
2. 'Try to find the fundamentals'
3. 'Try first to solve only a special case'
---
---
16. Repetition
1. Solve many problems
1. Drill
1. Copy
Repeat 0 or more repetitions of a similar
structure
+-------<------+
| |
-->-+->--[copy]-->-+->--
---
---
17. Reuse
1. Use existing solutions
1. 'Adapt existing solution to your needs'
1. Computer
1. Copy / Paste existing solution
---
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18. Search:
1. Consult external long term memory
Readily available, existing, solutions:
1. Books (=collective long term memory)
1. Dictionary (=collective long term memory)
1. Search for the origin of a word
2. Encyclopaedia (=collective long term memory)
3. Other
2. Internet
1. Search engine (=collective long term memory)
2. Newsgroups (=collective long term memory)
2. non readily available solutions:
1. Check all possibilities
1. Combinatorial search
1. Exhaustive search
Follow all possible paths to see which one leads most
expediently to the goal. This is an algorithmic solution
method, that is there is a guaranteed solution.
1. Brute force
Often just too many possible solutions, the exponential
or combinatorial explosion of possibilities.
1. Generate and test
1. 'I just simply test all possibilities, until
I find a solution'
1. Variation and selection
'I just simply test all possibilities, by
varying the current situation a little,
and then looking at the outcome. If
I have enough time, that might work'
2. not check all possibilities
1. Reduction
1. Difference-Reduction Method
Reduction of the distance between current state and
solution
1. Hill climbing
1. 'How can I choose those solutions which make the
distance between the current situation and the
goal smaller'
2. not reduction
1. Backtracking
1. 'How can I systematically test all or most of
the subpossibilities, and possibly skip some'
2. Backward search heuristic:
1. Start at the goal and search backwards for the
initial state.
2. Works best if there are less paths connected
to the goal state than to the initial states.
---
---
19. View:
1. Switch between generalization and specialization
1. Use a mixed scanning strategy
A mixed scanning strategy alternates a broad look at the
entire problem with in-depth looks at small parts of the
problem.
2. Generalization
1. Abstraction
1. 'What are the common factors in the examples I have
seen'
2. To generalize, replace the common words by words of a
more general class
(e.g. 'I eat apples' becomes
'I eat fruit', becomes
'I eat food')
2. Deduction
1. 'How can I apply this general method to this special
case'
3. Overview
1. Choosing a good representation for problem solving
A major determinant of the relative ease of solving a
problem is how the problem is represented
1. Choose a one to one mapping
2. Switch between different representations if
necessary
2. Graphical
1. 'Can I draw a picture'
1. 'Can I create some overview chart of the
problem'
1. Free hand
2. Non free hand
1. Hierarchical
1. Graph
1. Tree
2. Synthesis (=putting together)
Putting together various parts to arrange them into
something useful for the solution.
Selective combination.
Selective combination involves combining selectively
encoded information in such a way as to form a
logically integrated solution. Selective combination is
achieved by recognizing that the solution of a problem
consists of solving a set of nested problems, that is,
a number of subproblems, subproblems of the
subproblems, and so forth.
1. Put splitted parts together to a whole
1. Gestalt
1. The whole is more than the sum of the parts
Insight problems might require problem solvers
to perceive the problem as a whole
1. 'How can I get an overview, else I do not
see the wood through the trees'
2. Do not try to split parts
1. Try to work with the system as a whole
3. Get the overview
1. Draw pictures of the whole with links between
them
1. Draw a hierarchical tree
1. Work top down
2. Work from hypothesis to proof
3. Use working backwards from solution method
3. Specialization
1. Induction
1. Work from facts to conclusions
1. Use the working forwards to solution method
2. Work bottom up
1. 'Collect several examples, and try to see the
common factor'
2. Pattern recognition
1. 'Do you see a regular common pattern in all the
examples'
'Compare text 1
with text 2
with text 3
with text 4
...
with text last
and look for analogies, that is the 'same' or
'similar' text or pictures
---
---
Combinations of the above methods:
For example (often used in practice) use a combination of
analogy, pattern recognition, deduction and simplest example,
by comparing this similar examples in different books:
Recognize isomorphic structures:
Take book 1 and look for similar simplest example
Take book 2 and look for similar simplest example
Take book 3 and look for similar simplest example
Take book 4 and look for similar simplest example
...
Take book last and look for similar simplest example
-->-+->--[book1]-->-------+
| |
| |
+->--[book2]-->-------+-->--[find common structure]->-
| |
| |
+->--[book3]-->-------+
---
---
Note: some repetition, non-uniqueness and overlap is present in the
above tree. This because some of the methods have two or more
properties in common (e.g. and search and analogy, or and communication
and analogy, ...)
---
---
Note: some descriptions above might more fit the process of problem
solving, than the actual methods
---
---
Internet: see also:
---
Problem solving and creativity:
https://engineering.purdue.edu/ChE/News_and_Events/Publications/teachin
g_engineering/chapter5.pdf
---
problem-solving tests:
http://www.utexas.edu/student/utlc/handouts/1443.html
---
Thoughts on problem solving:
http://www.engin.umich.edu/~problemsolving/
---
Problem solving:
http://www.engr.mun.ca/~cdaley/1000/Design.html
---
Problem solving:
http://www.sparknotes.com/psychology/cognitive/problemsolving/section1.
html
---
Problem Solving Tutorial:
http://216.239.59.104/search?
q=cache:tPA2BA5paIwJ:isg.cs.tcd.ie/giangt/Tut_9.pdf+what+is+working+bac
kwards+problem+solving+method&hl=en&ie=UTF-8
---
Problem Solving Methods:
http://www.it.bton.ac.uk/staff/rng/teaching/notes/ProbSolvMethods.html#
Anderson
---
Creative problem solving techniques:
http://www.adbi.org/TM/by%20parts/APP01.pdf
---
Problem solving:
http://www.u.arizona.edu/~dusana/psych325presession/notes/CH11.ppt
---
Problem solving:
http://www.andrew.cmu.edu/course/85-211/ClassNotes/LN20.pdf
---
Development of expertise:
http://www2.sis.pitt.edu/~is2300dm/class-notes/09.html
---
Problem solving: Introduction to Computing for Engineers
http://www.cc.gatech.edu/classes/AY2004/cs1371_fall/lectures/CS1371_09_
Problem_Solving.ppt
---
Problem solving:
http://www.qca.org.uk/nq/ks/prob_level4.pdf
---
Problem solving research:
http://www.hawaii.edu/suremath/learn7.html
---
How to transform a novice in an expert problem solver:
http://web.mit.edu/tll/published/transforming_novice.htm
---
Problem solving (lateral thinking):
http://members.tripod.com/~eng50411/psolving.htm
---
Problem Solving Techniques:
http://www.virtualsalt.com/crebook4.htm
---
Problem solving:
http://www.unc.edu/~jmccabe/Ch10notes.html
---
Solution engineering:
http://home.att.net/~nickols/reengpsp.htm
---
Problem solving (Herbert Simon model):
http://www2.sis.pitt.edu/~is2300dm/class-notes/08.html
---
Problem solving (difference novice and expert problem solver)
http://www.phy.ilstu.edu/~wenning/ptefiles/311content/probsolving/exper
tnovice.html
---
Problem solving:
http://216.239.59.104/search?
q=cache:9nnyV7Y7fGsJ:homepage.usask.ca/~klo130/psych253/pdf/problem_sol
vingT.pdf+working+forward+problem+solving&hl=en&ie=UTF-8
---
Problem solving: Link: Overview: Can you give an overview of links?
http://www.faqts.com/knowledge_base/view.phtml/aid/33516/fid/1242
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